In a medium bowl or large measuring cup, whisk together the balsamic vinegar, honey, light brown sugar, 1 tablespoon of the olive oil, cornstarch, salt, and pepper until smooth; set the sauce aside.
Heat a large skillet over medium-high heat and add the remaining 2 tablespoons olive oil.
Add the thin-sliced chicken breasts to the skillet, season with salt and pepper, and cook about 5 minutes, flipping once, until the chicken is roughly 75% cooked through (timing will vary with thickness).
Pour the prepared balsamic sauce into the skillet; it may bubble briefly.
Arrange the broccoli florets and sugar snap peas evenly over and around the chicken. If needed, add 2 to 4 tablespoons water to create steam so the vegetables become crisp-tender.
Cover the skillet and steam for 3 to 5 minutes, or until the chicken is cooked through and the vegetables are crisp-tender.
Uncover, stir gently to coat the vegetables and chicken with the sauce, taste, and adjust seasoning with more salt, pepper, honey, or balsamic if desired before serving.
